A233090 Decimal expansion of Sum_{n>=1} (-1)^(n-1)*H(n)/n^2, where H(n) is the n-th harmonic number.

7, 5, 1, 2, 8, 5, 5, 6, 4, 4, 7, 4, 7, 4, 6, 4, 2, 8, 3, 7, 4, 8, 3, 6, 3, 5, 0, 9, 4, 4, 6, 5, 6, 2, 4, 4, 2, 2, 8, 1, 1, 6, 4, 3, 2, 7, 1, 2, 8, 1, 1, 8, 0, 1, 1, 2, 0, 1, 6, 9, 7, 2, 2, 0, 8, 8, 6, 4, 8, 8, 7, 8, 6, 1, 6, 4, 4, 5, 6, 8, 1, 3, 6, 6, 5, 3, 4, 9, 2, 1, 0, 0, 5, 8, 3, 4, 5, 3, 6, 3

Offset

0,1

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 43.

Ali Shadhar Olaikhan, An Introduction to the Harmonic Series and Logarithmic Integrals, 2021, p. 253, eq. (4.163).

Links

Table of n, a(n) for n=0..99.

R. Barbieri, J. A. Mignaco, and E. Remiddi, Electron form factors up to fourth order. I., Il Nuovo Cim. 11A (4) (1972), 824-864, table II (13).

Philippe Flajolet and Bruno Salvy, Euler Sums and Contour Integral Representations, Experimental Mathematics 7:1 (1998), pp. 15-35. See page 32.

Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, 2nd ed., Springer, 2020, p. 225, (C5.15).

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 561.

Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer International Publishing, 2019, section 4.53, p. 310, eq. (4.88), section 5.53, p. 327, section 6.52, pp. 508-513.

Index entries for sequences related to harmonic numbers.

Formula

Equals 5*zeta(3)/8.

Equals -Integral_{x=0..1} (log(1+x)*log(1-x)/x)*dx. - Amiram Eldar, May 06 2023

Equals Sum_{m>=1} Sum_{n>=1} (-1)^(m-1)/(m*n*(m + n)) (see Finch). - Stefano Spezia, Nov 02 2024

Equals Sum_{k>=1} H(k,2) / (k * 2^k), where H(k,2) = A007406(k)/A007407(k) is the k-th harmonic number of order 2 (Olaikhan, 2021). - Amiram Eldar, Feb 03 2026

Example

0.7512855644747464283748363509446562442281164327128118011201697220886...

Mathematica

RealDigits[ 5*Zeta[3]/8, 10, 100] // First

Prog

(PARI) 5*zeta(3)/8 \\ Amiram Eldar, Jun 04 2026

Crossrefs

Cf. A002117 (zeta(3)), A197070 (3*zeta(3)/4), A233091 (7*zeta(3)/8), A076788 (alternating sum with denominator n), A152648 (non-alternating sum with denominator n^2), A152649 (non-alternating sum with denominator n^3), A233033 (alternating sum with denominator n^3).

Cf. A007406, A007407.

Sequence in context: A093205 A156536 A110191 * A254177 A377609 A021575

Adjacent sequences: A233087 A233088 A233089 * A233091 A233092 A233093

Keyword

nonn,cons,changed

Author

Jean-François Alcover, Dec 04 2013, after the comment by Peter Bala about A233033

Status

approved